Satisfying Dose-Volume Constraints in Linear Fluence Map Optimization for IMPT

Journal of Cancer Therapy, 2014, 5, 198-207 Published Online February 2014 (http://www.scirp.org/journal/jct) http://dx.doi.org/10.4236/jct.2014.52025

An Automatic Approach for Satisfying Dose-Volume Constraints in Linear Fluence Map Optimization for IMPT

Maryam Zaghian1, Gino Lim1*, Wei Liu2, Radhe Mohan3

1Department of Industrial Engineering, University of Houston, Houston, USA; 2Department of Radiation Oncology, Mayo Clinic, Phoenix, USA; 3Department of Radiation Physics, The University of Texas MD Anderson Cancer Center, Houston, USA. Email: *[email protected]

Received December 4th, 2013; revised January 4th, 2014; accepted January 12th, 2014

Copyright © 2014 Maryam Zaghian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accor- dance of the Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the owner of the intellectual property Maryam Zaghian et al. All Copyright © 2014 are guarded by law and by SCIRP as a guardian.

ABSTRACT Prescriptions for radiation therapy are given in terms of dose-volume constraints (DVCs). Solving the fluence map optimization (FMO) problem while satisfying DVCs often requires a tedious trial-and-error for selecting appropriate dose control parameters on various organs. In this paper, we propose an iterative approach to satisfy DVCs using a multi-objective linear programming (LP) model for solving beamlet intensities. This algorithm, starting from arbitrary initial parameter values, gradually updates the values through an iterative solution process toward optimal solution. This method finds appropriate parameter values through the trade-off between OAR sparing and target coverage to improve the solution. We compared the plan quality and the satisfaction of the DVCs by the proposed algorithm with two nonlinear approaches: a nonlinear FMO model solved by using the L-BFGS algorithm and another approach solved by a commercial treatment planning system (Eclipse 8.9). We retrospectively selected from our institutional database five patients with lung cancer and one patient with prostate cancer for this study. Numerical results show that our approach successfully improved target coverage to meet the DVCs, while trying to keep corresponding OAR DVCs satisfied. The LBFGS algorithm for solving the nonlinear FMO model successfully satisfied the DVCs in three out of five test cases. However, there is no re- course in the nonlinear FMO model for correcting unsatisfied DVCs other than manually changing some parameter values through trial and error to derive a solution that more closely meets the DVC requirements. The LP-based heuristic algorithm outperformed the current treatment planning system in terms of DVC satisfaction. A major strength of the LP-based heuristic approach is that it is not sensitive to the starting condition.

KEYWORDS Fluence Map Optimization (FMO); Linear Programming (LP); Nonlinear Programming (NLP); Dose-Volume Constraint (DVC); Intensity-Modulated Proton Therapy (IMPT)

1. Introduction ment plans for IMPT, different inverse planning approaches to fluence map optimization (FMO) have been The primary goal of radiation therapy is to deliver the proposed [1]. prescribed dose to the target while sparing the adjacent Radiation oncologists use dose-volume constraints organs at risk (OARs) as much as possible. Intensity- (DVCs) to prescribe and control the dose to the target modulated proton therapy (IMPT) is a powerful tool for and OARs. The DVC specifies what fraction of a struc- designing and efficiently delivering highly conformal ture is allowed to receive a radiation dose higher than the dose distributions to the target while simultaneously specified upper threshold value or lower than the speci- sparing the neighboring OARs to a greater degree than fied lower threshold value. For example, according to the intensity-modulated radiation therapy. To optimize treat- lung protocol at The University of Texas MD Anderson *Corresponding author. Cancer Center, the treatment planner may specify that

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“no more than 37% of the normal lung is allowed to re- greedy algorithm has been applied to approximately ceive 20 Gy or more” instead of imposing a strict upper solve the optimization model faster than other existing dose limit of 20 Gy on the normal lung. Sometimes the algorithms. clinical goal of achieving an effective dose for all targets Nonlinear methods may provide local optimal or sub- while preserving the OARs cannot be met. In such cases, optimal solutions once DVCs are incorporated [4,14]. a compromise must be found and the DVCs relaxed. However, there is not sufficient information about the However, modifying DVCs is a highly complex problem, optimality gap to allow us to understand the difference and finding the global optimal solution can be very dif- between these local solutions and a global optimal solu- ficult [2,3]. tion. Romeijn, Ahuja, Dempsey, Kumar, and Li [15] ap- Linear programming (LP) is a powerful method for proximated any convex objective function by using a modeling FMO, but DVCs cannot be incorporated di- piecewise linear convex objective function that can be rectly into the FMO model without introducing integer solved quickly and easily for a global optimal solution. variables. Integer variables are needed because a DVC They incorporated an approximation of DVCs by formu- limits the dose applied for a certain number of voxels. lating conditional value-at-risk constraints on the diffe- Determining exactly how many of the voxels should rential dose-volume histogram (DVH). Langer and Leong meet DVCs is a difficult combinatorial problem that has [16] also formulated and solved the problem of optimiz- multiple local optima and is nonconvex and nondetermi- ing the beam weights under DVCs using linear program- nistic polynomial time hard [4,5]. ming. There are many formulations and solution methods for Romeijn, Ahuja, Dempsey, and Kumar [17] approx- the FMO problem under DVCs. Ehrgott, Güler, Ha- imated the DVC by “mean-tail-dose”, which refers to the macher, and Shao [6] reviewed the mathematical opti- mean dose of either the hottest or coldest specified vo- mization in intensity-modulated radiation therapy, in- lume. An advantage of the mean-tail-dose approach is cluding DVC-based models. One of the common nonli- that the metrics can be formulated linearly. Also, the near approaches minimizes the total weighted nonlinear global optimum can be more easily achieved because the function of the dose deviation violation [7,8]. Local problem is convex; however, DVC cannot be replaced by search methods have been reported to solve these models, a mean-tail-dose in the clinic. including gradient-based methods [8] and metaheuristics In the work by Chen, Herman, and Censor [2], two such as simulated annealing [9] and genetic algorithm existing approaches to satisfy DVCs were discussed: [10]. linear programming and ART3+, an adaptation of a pro- Xing, Li, Donaldson, Le, and Boyer [11] defined a jection method for solving feasible systems of linear in- DVH score function, and developed an algorithm that equalities. The two methods were compared in terms of performs a sequence of nonlinear optimizations which their ability to find an optimal solution as well as their updated the optimization parameters to improve the score. computational speed. Cho et al. [12] discussed two optimization techniques for Mixed-integer programming (MIP) is another tech- intensity beam modulation with DVCs. The first method, nique commonly used by researchers to model DVCs cost function minimization, applies a volume-sensitive [18-24]. However, MIP models are too difficult to solve penalty function in which fast simulated annealing is to be practically useful because of their nonconvex and used. In the second one, the convex projection method, nondeterministic polynomial time hard characteristics. the DVC is replaced by a limit on the integral dose. Tuncel, Preciado, Rardin, Langer, and Richard [5] intro- Michalski, Xiao, Censor, and Galvin [13] formulated a duced a family of disjunctive valid inequalities to the DVC satisfaction search for the discretized radiation MIP formulation of the FMO problem under DVCs. A therapy model. The aperture-based approach with prede- heuristic algorithm based on the geometric distance sort- fined segmental fields was used in inverse treatment ing technique is proposed by Lan, Li, Ren, Zhang, and planning, followed by an iterative algorithm of the si- Min [21] for solving a Linear constrained, quadratic ob- multaneous sub-gradient projections to obtain solutions. jective MIP formulation of the FMO with DVCs. Another model used to solve the FMO problem is the Preciado-Walters, Rardin, Langer, and Thai [22] for- weighted least-squares model, which focuses on dose- mulated the FMO problem as an MIP model over a volume-based weighted least-squares. These models are coupled pair of column generation processes. One of the fast, but they only give an approximate solution. Al- processes makes intensity maps, and the other determines though the DVC can be directly incorporated into the the protected area for organs under DVCs. Dink et al. objective function, the objective function will no longer [25,26] also incorporated DVCs into an MIP model: on be convex and differentiable. Zhang and Merritt [3] pre- the basis of work by Morrill, Lane, Wong, and Rosen sented a new least-squares model that has a monotonic [27]. and differentiable objective function. In their model, the Typically, the FMO problem that considers DVCs has

OPEN ACCESS JCT 200 An Automatic Approach for Satisfying Dose-Volume Constraints in Linear Fluence Map Optimization for IMPT multiple local optima and is non-deterministic polynomi- dose constraints used in the lung irradiation case were as al time hard [4,5]. There is no guarantee that the gra- follows: dient-based solution methods and metaheuristics pro- 1) 3ODQQLQJWDUJHWYROXPH 379 RIWKH379 posed to solve non-LP formulations would allow us to YROXPHUHFHLYHVRIWKHSUHVFULEHGGRVH find a global optimal solution. Those methods usually 2) PTV: No more than 2 cm3 UHFHLYHVRIWKH find local optimal and/or suboptimal solutions [5]. In prescribed dose (minor deviation) or no more than 2 cm3 contrast, MIP models can guarantee global optimality. UHFHLYHVRIWKHSUHVFULEHGGRVH QRGHYLDWLRQ . But solving the MIP models with DVCs can take such a 3) 1RUPDOOXQJ9. long time that they may not be useful in clinical practice 4) Normal lung: Mean lung GRVHGy relative bio- [14]. logical effectiveness. Multiobjective optimization is often used to optimize 5) +HDUW9. fluence maps, although choosing the appropriate values 6) +HDUWPHDQGRVHGy relative biological effec- for the weight parameters and hot- and cold-spot control tiveness. parameters is not straightforward and requires a trial-and-error approach. In this paper, we developed a Li- 2.3. Linear FMO near-based heuristic algorithm to satisfy DVCs that eli- The FMO model optimizes the amount of radiation that minates the manual selection of those parameters. This algorithm begins by setting loose boundaries for hot- and each beamlet delivers when the gantry is positioned at a cold-spot control parameters; then, after checking the given angle. The goal of this model is to find the optimal dose-volume criteria, it gradually tightens the boundaries, beamlet weights, assuming that a set of beam angles are if the DVC criteria are not satisfied. At the same time, given as input parameters. The objective function for this our algorithm increases the objective function weight model can be either linear or nonlinear. We describe our parameters for organs but does not satisfy their corres- model, which is based on the linear FMO model by Lim, ponding DVCs. Although this proposed method should Choi, and Mohan [28] in Section 2.4. Then, in Section work well for intensity-modulated radiation therapy, our 2.5, a nonlinear alternative FMO is presented. The input method was developed specifically for IMPT. Proton parameters for the linear FMO are shown in Table 2. A therapy is increasingly being used to treat cancer patients cold spot represents a portion of a structure that receives in clinical practice. Thus, we tested our DVC satisfaction less than the desired radiation dose. A hot spot represents algorithm on IMPT cases. a portion of a structure that receives a dose higher than the desired upper boundary. 2. Materials and Methods The voxels in the OARs are denoted by Sk to signify a 2.1. Patient Data and Beam Configurations collection of k OARs, and T represents the voxels in the PTV. Each voxel is represented by a three dimensional We evaluated the relative performance of the optimiza- coordinate, (x,y,z). The dose contribution d(x,y,z,j) is tion approaches by retrospectively creating treatment calculated via an in-house-developed dose calculation plans for one patient with prostate cancer and five pa- engine [29], where d(x,y,z,j) denotes the dose contributed tients with lung cancer who had previously undergone by the jth beamlet per unit weight, and is received by IMPT on a prospective institutional review board-ap- voxel (x,y,z). Given that the decision variable wj is the proved protocol at MD Anderson Cancer Center in Hou- intensity of beamlet j, the total dose in voxel (x,y,z) can ston, Texas. Two lateral beam fields were used for the m be calculated as Dwd . prostate case while three beam fields were used for the xyz,, ¦ j 1 j xyzj,,, lung cancer cases. The prescribed doses and beam con- The standard LP model to optimize the beamlet figurations are listed in Table 1. For the patients with weights is constructed as follows. The objective function lung cancer, which are more complicated than the patient of the LP model has three terms that are associated with with prostate cancer, we used two methods of FMO (li- the target and OARs. near and nonlinear) to evaluate DVC satisfaction. OTtTUT De OTtLTT eD min fD 11 2.2. DVC w TT In this study, we used the same DVCs used in the origi- OMSSSS De kkkk 1 nal treatment protocols and adopted in the clinic. The ¦ st.. LTTTdd D U following DVC protocols were used for the prostate k Sk cancer case: UHFWXP9 RIWKHUHFWXP where, m for all (x,y,z). Note Dwd xyz,, ¦ j 1 j xyzj,,, is allowed to receive 70 Gy or more); 2) bladder V65 that for a vector x, 1-norm x is the sum of the ab- 25%; and EODGGHU9 The DVCs and mean- 1

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Table 1. Dose and beam configurations. ues are determined, which results in an even trade-off between OAR sparing and target coverage. The algo- Target Prescribed Gantry Dose grid Patient volume rithm stops when all of the DVCs are satisfied or when dose (Gy) angles resolution (mm) (cm3 ) no more improvement can be made in the DVCs. The Prostate 76 99 90, 270 4 algorithm can be described as follows: Lung 1 74 922 320, 230, 180 4 1) Initialize ĳ and ȜS for OARs, as well as Ot , Ot , șL, Lung 2 60 616 100, 150, 190 5 and șU for the target. 2) Solve the FMO model. Lung 3 66 800 200, 225, 300 5 3) If all constraints are satisfied, stop; otherwise go to Lung 4 74 407 225, 205,160 5 step 4. Lung 5 74 502 200, 90, 150 5 4) Remove all OAR voxels that satisfy the DVCs from sets S and keep the remaining (or DVC-violating) vox- Table 2. Input parameters for linear FMO. k els for the next iteration, if there are any violated DVCs Parameter Definition for OARs. T A set of voxels in planning target volume (PTV or target) 5) Decrease the hot spot control parameter (ĳ) and in- th crease the penalty coefficient for hot spots (ȜS) if any of Sk A set of voxels in k organ-at-risk (OAR) ș Cold spot control parameter on PTV the constraints of an OAR are not satisfied. L 6) For violated DVCs of the target, increase the cor- ș Hot spot control parameter on PTV U responding penalty coefficient ( O and O ), tighten the ĳ Hot spot control parameter on OAR t t corresponding control parameter (increase șL or decrease LT Lower reference boundary on PTV șU), and then go to step 2. UT Upper reference boundary on PTV Ot Penalty coefficient for hot spots on PTV 2.5. Nonlinear FMO O Penalty coefficient for cold spots on PTV t This problem can also be modeled using the nonlinear th ȜSk Penalty coefficient for hot spots on k OAR (mostly quadratic) objective function. A quadratic mo- Dose contribution from beamlet j to voxel (x,y,z) d(x,y,z,j) del used to optimize the weights is constructed as fol- Where (x,y,z) T S and j {1, 2, ..., m} lows: 22 §· fD OOtTP D D tPT D D solute values of the columns ¨¸¦ xi . The notation (.)+ 22 i ©¹ 2 represents max{•,0}, DT and DS are doses to the target OSDD S Tol . ¦ kk k k 2 and OARs respectively, and eT and eS are the vectors of ones. Similar to the LP model, Ȝ terms denote the penalty weights of the corresponding organs. DP and DP are 2.4. LP Heuristic Algorithm to Satisfy DVCs in the upper and lower prescribed dose levels for the tumor, Linear FMO respectively, and D is the tolerance dose for the kth Tolk The linear FMO model [28] imposes strict hot spot con- OAR. trol parameters on both the target and the OARs and In order to take in to account the non-negativity of the beamlet intensities, the weight of beamlet j is denoted by strict cold spot control parameters on the target. This 2 approach often helps satisfy the DVCs if the values are the non-negative quantity wj . As a result, the con- selected appropriately. The values of the parameters can strained optimization problem with respect to weights is sometimes be shown to be unnecessarily conservative, turned into an unconstrained one of optimizing the which compromises the quality of the treatment plan for square root of the beamlet weights instead of optimizing other organs. Consequently, a tedious trial-and-error ef- the beamlet weight directly. fort is made to find appropriate parameter values. To The objective function penalizes quadratic integral eliminate this trial-and-error approach, applying the dose violations to the target and the OARs. The model techniques for controlling dose-volume histogram by only includes OAR voxels receiving doses greater than Lim, Ferris, Shepard, Wright, and Earl [30], our algo- their tolerance dose levels. The penalty weights and the rithm starts by setting arbitrarily loose upper boundaries prescribed and tolerance doses for the target and OARs on these parameters; it then gradually tightens the boun- can be altered by the treatment planner. The limited daries and simultaneously increases the objective penalty memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) coefficients through an iterative solution process. Fol- method is applied to solve this unconstrained optimiza- lowing this iterative process, appropriate parameter val- tion model [31,32].

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3. Results 3.1. Prostate Cancer Case We tested our LP heuristic algorithm on a prostate cancer case. Two major OARs were the rectum and bladder. Note that we normalized the prescribed dose on the PTV so that the value of 1 stands for the prescribed radiation dosage level on the tumor. The normalization was done for all the experiments in this paper. Typically, the values of Ot , Ot , and ȜS are selected by the treatment planner. In our experiments, we started the algorithm by assigning equal weights to the PTV and OARs. The algorithm then adjusted these values in order to satisfy the DVCs. For the prostate cancer case, all DVCs were satisfied by the algorithm at its initial iteration (which is the same as the linear FMO) and the algorithm was terminated.

However, to assess the algorithm, we modified the DVCs to make them much harder to satisfy. Figure 1 shows Figure 1. Effect of the LP heuristic on the dose-volume histogram for the prostate case (solid line: initial iteration; how using the algorithm for organ sparing and to main- dashed line: final iteration). tain the target dose coverage affected the DVH. are displayed in Figure 2 for all five cases of lung cancer. 3.2. Lung Cancer Cases DVCs 1, 3, and 5 are illustrated for all the cases. For Pa- 3.2.1. Results for the LP Heuristic Approach tient 2, the algorithm was not able to satisfy DVC3, but In this section, we applied our developed method to five tried to improve tumor coverage while keeping the total cases of lung cancer. These cases were more complicated lung’s DVH as close to the corresponding DVC as possi- than the case of the patient with prostate cancer discussed ble. DVC2 was obviously satisfied, since for all the cases, in the previous section. Lung cancer cases include two of no volume of the PTV received a dose greater than or the most critical OARs: normal lung and heart. Table 3 equal to 120% of the prescribed dose. Mean dose con- presents the progression of the LP heuristic algorithm, straints (4 and 6) which could not be displayed on the from the initial iteration to the final one in terms of con- DVH were also satisfied for all the patients (Table 3). straints satisfaction for all the lung cancer cases. The The outcomes of our constrained FMO model de- results of the first iteration showed unnecessarily con- pended on the lower and upper reference boundary pa- servative OAR sparing, whereas the PTV coverage was rameters for PTV, LT and UT, respectively. Having those not enough to satisfy its corresponding DVCs. parameters in the LP model is useful in controlling the For instance, DVC1 at the first iteration was not satis- optimization process, which is not possible in uncon- fied for Patient 1. As illustrated in Table 3, 80.7% of the strained optimization models. However, selecting the right PTV received a dose greater than or equal to 95% of the parameter values can be a daunting task. Choosing boun- prescribed dose, which was not greater than or equal to daries that are too tight can result in an infeasible solu- the corresponding reference point (95%) and thus vi- tion, so we imposed loose boundary constraints on the olated the reference criteria. On the other hand, for target, i.e., LT = 0.8 and UT = 1.2, for all cases. Further DVC3, 21.6% of the normal lung received a dose greater experiments were performed to show the behavior of the than or equal to 20 Gy at the first iteration, which was model. Four different settings of these boundaries were significantly less than the reference point (37%). How- implemented for Patient 4, as shown in Figure 3. The ever, at the final iteration of the LP heuristic for the same cold spots were controlled by setting different values of patient, 99.6% of the PTV received a dose greater than or LT. When we increased the lower boundary from 0.8 equal to 95% of the prescribed dose, which satisfied the (Figure 3(a)) to 0.85 (Figure 3(c)), the model respon- reference criteria for DVC1, and 29.7% of the normal ded accordingly and the starting point of the target DVH lung received a dose greater than or equal to 20 Gy, curve has been shifted to ensure that no target voxels which was still less than the reference point for DVC3 received less than 85% of the prescribed dose. Simi- (37%). larly, the hot spots can be controlled by UT, as seen in Hence, in the last iteration, the LP heuristic found so- Figures 3(a) and (b). Figure 3(b) is the result of allow- lutions that satisfied the constraints for all organs through ing up to a 20% overdose (UT = 1.2) to the target, an iterative process by automatically adjusting the opti- while Figure 3(a) is based on a 15% overdose limit, i.e., mization model parameter values. The resulting DVHs UT = 1.15. By setting these boundaries tighter, both the

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(a) (b)

(e) Figure 2. Dose-volume histograms using the LP heuristic algorithm for cases of lung cancer (blue line: target, red line: normal lung, black line: heart). (a): Patient 1; (b): Patient 2; (c): Patient 3; (d): Patient 4; (e): Patient 5.

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Table 3. Comparison of constraints satisfaction levelsa between the initial and final iterations of the LP heuristic in lung cancer cases.

Reference Patient 1 Patient 2 Patient 3 Patient 4 Patient 5 Const. criteria First itr. Final itr. First itr. Final itr. First itr. Final itr. First itr. Final itr. First itr. Final itr. 1 0.807 0.996 0.706 1.000 0.785 0.997 0.820 0.958 0.852 1.000 2 FP3b 0 0 0 0 0 0 0 0 0 0 3 0.216 0.297 0.238 0.385 0.230 0.294 0.147 0.180 0.106 0.218 4 Gy[RBE] 13.47 18.13 12.12 19.32 11.48 15.18 9.47 11.47 7.99 12.728 5 0.084 0.102 0.089 0.117 0.060 0.089 0.028 0.043 0.009 0.007 6 Gy[RBE] 6.81 8.806 8.94 10.26 5.74 7.656 2.88 4.14 1.92 1.332

Abbreviations: Const., constraint; RBE, relative biological effectiveness; itr., iteration. aPortion of voxels satisfied the constraint. bMinor deviation was allowed for DVC2.

(a) (b)

(c) (d) Figure 3. Dose-volume histogram using the LP heuristic for four reference boundary settings on Patient 4 (blue line: target, red line: normal lung, black line: heart; Blue star: DVC1, red star: DVC3, black star: DVC5). (a): LT = 0.8, UT = 1.2; (b): LT = 0.8, UT = 1.15; (c): LT = 0.85, UT = 1.2; (d): LT = 0.85, UT = 1.15.

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cold spots and the hot spots were reduced accordingly. Note that such a reduction trend will continue as long as the model can find a feasible solution to meet the constraints requirements. If the model cannot find a solution to satisfy the protocol requirements, the algorithm termina- tes with the last solution found during the iterative process.

3.2.2. Comparison between the LP Heuristic and the NLP Model We applied the nonlinear model solved by L-BFGS to the lung cancer cases and compared the results to those of the LP heuristic approach. Although the NLP model was able to satisfy constraints for some cases, it failed for other cases. Figure 4 compares DVHs of the LP heuristic and NLP approaches for one of these cases. All constraints except DVC1 were satisfied by the NLP approach on this case (Patient 4). Note that DVC1 corres- ponds to cold spots on the tumor; 91% of the tumor vo- Figure 4. Comparison of dose-volume histograms using the lume received a dose greater than or equal to 95% of the LP heuristic and NLP approach for Patient 4 (solid line: NLP, dashed line: LP heuristic;blue line: target, red line: prescribed dose, which was less than the reference point total lung, black line: heart;blue star: DVC1, red star: on DVC1 (95%) and violated the reference criteria. DVC3, black star: DVC5). When compared to the NLP FMO, the LP heuristic resulted in fewer cold spots and hot spots on the tumor. 4. Discussion and Conclusions The worst cold spot on the tumor using the NLP FMO was 72% of the prescribed dose, while the worst cold The purpose of this study was to test an LP-based heuris- spot using the LP heuristic was 80% of the prescribed tic approach for FMO with DVCs using an iterative li- dose. The worst hot spot on the tumor using the NLP near FMO algorithm. A major advantage of our method FMO was more than 120% of the prescribed dose, while is that it satisfies the DVCs without increasing the com- the worst hot spot using the LP heuristic was 112% of the plexity of the problem and while conserving its convexi- ty. This method alleviates the tedious effort of selecting prescribed dose. initial values of model parameters by choosing appropriate values through a simple iterative process. Starting 3.2.3. Comparison of LP Heuristic with Eclipse 8.9 from its initial condition, the parameter values are itera- For the five cases of lung cancer, we compared the out- tively updated while considering trade-offs between the comes of our method to the plans from Eclipse 8.9 com- target coverage and the OAR sparing. mercial treatment planning system (Varian Medical Sys- To validate the outcomes of our new algorithm, the tems, Palo Alto, CA, USA), which uses an NLP solver results for cases of lung cancer were compared with and is used at MD Anderson to treat cancer patients. those of two other approaches (NLP FMO solved by Those plans are physician-approved clinical ones gener- L-BFGS and Eclipse 8.9). We first illustrated the LP heu- ated by very experienced dosimetrists, and satisfy all ristic algorithm on a case of prostate cancer (Figure 1). clinical requirements. Table 4 presents the satisfaction Using the iterative process, the algorithm found a better levels for all constraints using our heuristic algorithm solution in terms of satisfying the constraints on all and the Eclipse 8.9 system. For all patients, the LP heu- OARs. We then examined the performance of all three ristic algorithm satisfied all the constraints except DVC 3 solution methods in the five lung cancer cases. We ob- for Patient 2. However, using the commercial treatment served the progression of the iterative LP heuristic when planning system resulted in more violations from the comparing the outcomes of the first and the last iteration reference points: Constraints 2, 3, 4, and 5 for Patient 1, (Table 3). DVCs corresponding to cold spots on the tu- constraints 3 and 4 for Patient 2, and constraints 1 and 4 mor were not satisfied at the initial iteration for any of for Patient 3 (see Table 4). As an example, for Patient 3, the five cases. On the other hand, the satisfied portion of 94.5% of the PTV volume received a dose greater than or constraints corresponding to total lung and heart were equal to 95% of the prescribed dose, which was slightly much higher than the acceptable level (reference point). below the reference point for DVC1 (95%). Also, the Thus, our heuristic algorithm improved the target cover- mean lung dose for this patient was 24.11 Gy, which vi- age step-by-step to meet the DVC while maintaining olated the reference criteria corresponding to constraint 4 satisfaction of the constraints for the OARs. We illu- (20 Gy). strated how to use the LP heuristic approach to control

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Table 4. Comparison of constraints satisfaction levels* between LP heuristic and Eclipse 8.9.

Patient1 Patient2 Patient3 Patient4 Patient5 Const. Referencecriteria Eclipse LP heu. Eclipse LP heu. Eclipse LP heu. Eclipse LP heu. Eclipse LP heu. 1 0.993 0.996 0.983 1.000 0.945 0.997 0.956 0.958 0.98 1.000 2 FP3** 5.26 0 0 0 0 0 0 0 0 0 3 0.7 0.297 0.408 0.385 0.271 0.294 0.185 0.180 0.224 0.218 4 20 Gy[RBE] 25.01 18.13 32.76 19.32 24.11 15.18 16.52 11.47 13.02 12.728 5 0.3 0.4 0.102 0.170 0.117 0.085 0.089 0.044 0.043 0.025 0.007 6 35Gy[RBE] 14.98 8.806 25.11 10.26 13.33 7.656 6.68 4.14 4.366 1.332 Abbreviations: Const., constraint; RBE, relative biological effectiveness; itr., iteration; LP heu., LP heuristic. *Portion of voxels satisfied the constraint. **Minor deviation was allowed for DVC2. both the hot spots and the cold spots, using four different [2] W. Chen, G. Herman and Y. Censor, “Algorithms for settings of reference boundaries on the tumor (Figure 3). Satisfying Dose Volume Constraints in Intensity-Modu- We showed that the LP heuristic successfully satisfied all lated Radiation Therapy,” Proceedings of the Interdiscip- linary Workshop on Mathematical Methods in Biomedical constraints in all cases and demonstrated that properly Imaging and Intensity-Modulated Radiation Therapy selected tighter boundaries decreased cold spots and hot (IMRT), Pisa, October 2007. spots on the tumor as expected. These outcomes showed [3] Y. Zhang and M. Merritt, “Dose-Volume-Based IMRT the advantage of using constrained linear programming Fluence Optimization: A Fast Least-Squares Approach optimization to control the cold and hot spots on the tu- with Differentiability,” Linear Algebra and Its Applica- mor. tions, Vol. 428, No. 5, 2008, pp. 1365-1387. We also compared the performance of the LP heuristic http://dx.doi.org/10.1016/j.laa.2007.09.037 approach and the NLP approach in lung cancer cases and [4] J. O. Deasy, “Multiple Local Minima in Radiotherapy observed that the NLP failed to satisfy the DVCs in some Optimization Problems with Dose-Volume Constraints,” cases (Figure 4). Note that the NLP model is an uncon- Medical Physics, Vol. 24, 1997, pp. 1157-1162. strained minimization model solved using the L-BFGS http://dx.doi.org/10.1118/1.598017 method. A major drawback of this approach is that the [5] A. T. Tuncel, F. Preciado, R. L. Rardin, M. Langer and J. model does not consider DVCs. Hence, there is no re- P. P. Richard, “Strong Valid Inequalities for Fluence Map Optimization Problem under Dose-Volume Restrictions,” course for correcting unsatisfied constraints other than a Annals of Operations Research, Vol. 196, No. 1, 2012, pp. manual trial-and-error effort to find a solution that is 819-840. http://dx.doi.org/10.1118/1.598017 close to the requirements. One can try different values of [6] M. Ehrgott, Ç. Güler, H. W. Hamacher and L. Shao, initial beamlet weights and prescribed dose, or different “Mathematical Optimization in Intensity Modulated Rad- penalty weights on the objective function, which requires iation Therapy,” 4OR, Vol. 6, No. 3, 2008, pp. 199-262. a substantial amount of effort to find a clinically feasible http://dx.doi.org/10.1118/1.598017 solution. The advantage of using the LP heuristic ap- [7] T. Bortfeld, J. Stein and K. Preiser, “Clinically Relevant proach is that it is not sensitive to its initial condition. It Intensity Modulation Optimization Using Physical Crite- may fail to satisfy the DVCs or may result in an unac- ria,” Proceedings of the XII ICCR, Salt Lake City, Utah, ceptable plan quality at the first iteration, but through the 27-30 May, 1997. iterative process, it will find a solution to meet the re- [8] Q. Wu and R. Mohan, “Algorithms and Functionality of quirements. an Intensity Modulated Radiotherapy Optimization Sys- tem,” Medical Physics, Vol. 27, 2000, pp. 701-711. The LP heuristic approach performed better than the http://dx.doi.org/10.1118/1.598932 Eclipse 8.9 commercial treatment planning system for [9] X. Wu, Y. Zhu, J. Dai and Z. Wang, “Selection and De- the five lung cancer cases (Table 4). Unlike Eclipse 8.9, termination of Beam Weights Based on Genetic Algo- the LP heuristic approach satisfied the constraints for rithms for Conformal Radiotherapy Treatment Planning,” almost all the reference criteria for all patients. The Ec- Physics in Medicine and Biology, Vol. 45, No. 9, 2000, lipse treatment planning system produced solutions in pp. 2547-2558. which eight constrains were violated in three cases. http://dx.doi.org/10.1088/0031-9155/45/9/308 [10] M. Langer, R. Brown, S. Morrill, R. Lane and O. Lee, “A Generic Genetic Algorithm for Generating Beam Wei- REFERENCES ghts,” Medical Physics, Vol. 23, 1996, pp. 965-972. [1] A. Lomax, “Intensity Modulation Methods for Proton [11] L. Xing, J. G. Li, S. Donaldson, Q. T. Le, and A. L. Boy- Radiotherapy,” Physics in Medicine and Biology, Vol. 44, er, “Optimization of Importance Factors in Inverse Plan- No. 1, 1999, pp. 185-205. ning,” Physics in Medicine and Biology, Vol. 44, 1999, http://dx.doi.org/10.1088/0031-9155/44/1/014 pp. 2525-2536.

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[12] P.S. Cho, S. Lee, R. J. Marks II, S. Oh, S. G. Sutlief and “A Coupled Column Generation, Mixed Integer Ap- M. H. Phillips, “Optimization of Intensity Modulated proach to Optimal Planning of Intensity Modulated Radi- Beams with Volume Constraints Using Two Methods: ation Therapy for Cancer,” Mathematical Programming, Cost Function Minimization and Projections onto Convex Vol. 101, No. 2, 2004, pp. 319-338. Sets,” Medical Physics, Vol. 25, 1998, pp. 435-444. http://dx.doi.org/10.1007/s10107-004-0527-6 http://dx.doi.org/10.1118/1.598218 [23] M. Langer, R. Brown, P. Kijewski and C. Ha, “The Re- [13] D. Michalski, Y. Xiao, Y. Censor, J. M. Galvin, “The liability of Optimization under Dose-Volume Limits,” Ma- Dose-Volume Constraint Satisfaction Problem for Inverse thematical Programming, Vol. 26, No. 3, 1993, pp. 529- Treatment Planning with Field Segments,” Physics in 538. http://dx.doi.org/10.1016/0360-3016(93)90972-X Medicine and Biology, Vol. 49, No. 4, 2004, pp. 601- [24] M. Langer, R. Brown, M. Urie, J. Leong, M. Stracher and 616. J. Shapiro, “Large Scale Optimization of Beam Weights [14] Q. Wu and R. Mohan, “Multiple Local Minima in IMRT under Dose-Volume Restrictions,” International Journal Optimization Based on Dose-Volume Criteria,” Medical of Radiation Oncology * Biology * Physics, Vol. 18, No. Physics, Vol. 29, 2002, pp. 1514-1528. 4, 1990, pp. 887-893. http://dx.doi.org/10.1118/1.1485059 http://dx.doi.org/10.1016/0360-3016(90)90413-E [15] H. E. Romeijn, R. K. Ahuja, J. F. Dempsey, A. Kumar [25] D. Dink, M. Langer, S. Orcun, J. Pekny, R. Rardin, G. and J. G. Li, “A Novel Linear Programming Approach to Reklaitis and B. Saka, “IMRT Optimization with Both Fluence Map Optimization for Intensity Modulated Radi- Fractionation and Cumulative Constraints,” American Jour- ation Therapy Treatment Planning,” Physics in Medicine nal of Operations Research, Vol. 1, No. 3, 2011, pp. 160- and Biology, Vol. 48, No. 21, 2003, pp. 3521-3542. 171. http://dx.doi.org/10.4236/ajor.2011.13018 http://dx.doi.org/10.1088/0031-9155/48/21/005 [26] D. Dink, M. P. Langer, R. L. Rardin, J. F. Pekny, G. V. [16] M. Langer and J. Leong, “Optimization of Beam Weights Reklaitis and B. Saka, “Intensity Modulated Radiation under Dose-Volume Restrictions,” International Journal Therapy with Field Rotation—A Time-Varying Fractio- of Radiation Oncology * Biology * Physics, Vol. 13, No. nation Study,” Health Care Management Science, Vol. 15, 8, 1987, pp. 1255-1260. No. 2, 2012, pp. 138-154. http://dx.doi.org/10.1016/0360-3016(87)90203-3 http://dx.doi.org/10.4236/ajor.2011.13018 [17] H. E. Romeijn, R. K. Ahuja, J. F. Dempsey and A. Kumar, [27] S. M. Morrill, R. G. Lane, J. A. Wong, I. I. Rosen, “A New Linear Programming Approach to Radiation The- “Dose-Volume Considerations with Linear Programming rapy Treatment Planning Problems,” Operations Research, Optimization,” Medical Physics, Vol. 18, 1991, pp. 1201- Vol. 54, No. 2, 2006, pp. 201-216. 1211. http://dx.doi.org/10.1118/1.596592 http://dx.doi.org/10.1287/opre.1050.0261 [28] G. J. Lim, J. Choi and R. Mohan, “Iterative Solution Me- [18] H. Rocha, J. M. Dias, B. C. Ferreira and M. C. Lopes, thods for Beam Angle and Fluence Map Optimization in “Discretization of Optimal Beamlet Intensities in IMRT: Intensity Modulated Radiation Therapy Planning,” OR A Binary Integer Programming Approach,” Search Re- Spectrum, Vol. 30, No. 2, 2008, pp. 289-309. sultsMathematical and Computer Modelling, Vol. 428, http://dx.doi.org/10.1007/s00291-007-0096-1 No. 5, 2011, pp. 1345-1364. [29] Y. Li, R. X. Zhu, N. Sahoo, A. Anand and X. Zhang, [19] M. C. Ferris, R. R. Meyer and W. D’Souza, “Radiation “Beyond Gaussians: A Study of Single-Spot Modeling for Treatment Planning: Mixed Integer Programming For- Scanning Proton Dose Calculation,” Physics in Medicine mulations and Approaches,” Handbook on Modelling for and Biology, Vol. 57, No. 4, 2012, pp. 983-997. Discrete Optimization, 2006, pp. 317-340. http://dx.doi.org/10.1088/0031-9155/57/4/983 [20] E. K. Lee, T. Fox and I. Crocker, “Integer Programming [30] G. J. Lim, M. C. Ferris, D. M. Shepard, S. J. Wright and Applied to Intensity-Modulated Radiation Therapy Treat- M. A. Earl, “An Optimization Framework for Conformal ment Planning,” Annals of Operations Research, Vol. 119, Radiation Treatment Planning,” INFORMS Journal on No. 1, 2003, pp. 165-181. Computing, Vol. 19, No. 3, 2007, pp. 366-380. http://dx.doi.org/10.1023/A:1022938707934 http://dx.doi.org/10.1287/ijoc.1060.0179 [21] Y. Lan, C. Li, H. Ren, Y. Zhang and Z. Min, “Fluence [31] D. C. Liu and J. Nocedal, “On the Limited Memory Map Optimization (FMO) with Dose-Volume Constraints BFGS Method for Large Scale Optimization,” Mathe- in IMRT Using the Geometric Distance Sorting Method,” matical Programming, Vol. 45, No. 1, 1989, pp. 503-528. Physics in Medicine and Biology, Vol. 57, No. 20, 2012, http://dx.doi.org/10.1007/BF01589116 pp. 6407-6428. [32] S. Bochkanov and V. Bystritsky, “ALGLIB,” http://dx.doi.org/10.1088/0031-9155/57/20/6407 www.alglib.net [22] F. Preciado-Walters, R. Rardin, M. Langer and V. Thai,

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